# Polydisc

In the theory of functions of several complex variables, a branch of mathematics, a **polydisc** is a Cartesian product of discs.

More specifically, if we denote by the open disc of center *z* and radius *r* in the complex plane, then an open polydisc is a set of the form

It can be equivalently written as

One should not confuse the polydisc with the open ball in **C**^{n}, which is defined as

Here, the norm is the Euclidean distance in **C**^{n}.

When , open balls and open polydiscs are *not* biholomorphically equivalent, that is, there is no biholomorphic mapping between the two. This was proven by Poincaré in 1907 by showing that their automorphism groups have different dimensions as Lie groups.[1]

When the term *bidisc* is sometimes used.

A polydisc is an example of logarithmically convex Reinhardt domain.

## References

- Poincare, H,Les fonctions analytiques de deux variables et la r?epresentation conforme, Rend. Circ. Mat. Palermo23 (1907), 185-220

- Steven G Krantz (Jan 1, 2002).
*Function Theory of Several Complex Variables*. American Mathematical Society. ISBN 0-8218-2724-3. - John P D'Angelo, D'Angelo P D'Angelo (Jan 6, 1993).
*Several Complex Variables and the Geometry of Real Hypersurfaces*. CRC Press. ISBN 0-8493-8272-6.

*This article incorporates material from polydisc on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*